Can somebody explain the plate trick to me?

I learned of the plate trick via Wikipedia, which states that this is a demonstration of the fact that SU(2) double-covers SO(3). It also offers a link to an animation of the "belt trick" which is apparently equivalent to the plate trick. Since I've thought most about the belt version, I'll phrase my question in terms of the belt trick.

I am not clear on how the plate/belt trick relates to the double covering. Specifically, I am looking for a sort of translation of each step of the belt trick into the Lie group setting. For example, am I correct in interpreting the initial twisting of the belt as corresponding to the action of a point in SU(2)? Which point? Do I have the group right?

A 360 degree rotation of the arm results in a very sore arm unless you unwind it or wind it a second time. Think about the rotation of the hand as being a path in the space of all rotations of 3-space. Or if you really want to rotate something, tape a belt to a soccer ball and tape the other end of the belt to the wall. Just rotate the ball a full rotation, the belt gets a twist. Now rotate again; the belt has two full twists. The twists in the belt can be undone by looping the ball back through the belt as in the animation.

The diagram in the lower right shows the paths in the space $\operatorname{SO}(3)$ which is the $3$-ball with its antipodal points on its boundary identified. That is why the doubled path appears to be broken; it is passing through antipodal points.

You might play with a Möbius band and the pencil loop which maps twice around to gain more intuition. The point of thinking of the Möbius band is that it is embedded in the space of rotations.

It is really worth thinking this through while going to sleep or while riding the bus!